# :mod:decimal --- Decimal fixed point and floating point arithmetic

The :mod:decimal module provides support for fast correctly-rounded decimal floating point arithmetic. It offers several advantages over the :class:float datatype:

• Decimal "is based on a floating-point model which was designed with people in mind, and necessarily has a paramount guiding principle -- computers must provide an arithmetic that works in the same way as the arithmetic that people learn at school." -- excerpt from the decimal arithmetic specification.

• Decimal numbers can be represented exactly. In contrast, numbers like :const:1.1 and :const:2.2 do not have exact representations in binary floating point. End users typically would not expect 1.1 + 2.2 to display as :const:3.3000000000000003 as it does with binary floating point.

• The exactness carries over into arithmetic. In decimal floating point, 0.1 + 0.1 + 0.1 - 0.3 is exactly equal to zero. In binary floating point, the result is :const:5.5511151231257827e-017. While near to zero, the differences prevent reliable equality testing and differences can accumulate. For this reason, decimal is preferred in accounting applications which have strict equality invariants.

• The decimal module incorporates a notion of significant places so that 1.30 + 1.20 is :const:2.50. The trailing zero is kept to indicate significance. This is the customary presentation for monetary applications. For multiplication, the "schoolbook" approach uses all the figures in the multiplicands. For instance, 1.3 * 1.2 gives :const:1.56 while 1.30 * 1.20 gives :const:1.5600.

• Unlike hardware based binary floating point, the decimal module has a user alterable precision (defaulting to 28 places) which can be as large as needed for a given problem:

>>> from decimal import *
>>> getcontext().prec = 6
>>> Decimal(1) / Decimal(7)
Decimal('0.142857')
>>> getcontext().prec = 28
>>> Decimal(1) / Decimal(7)
Decimal('0.1428571428571428571428571429')

• Both binary and decimal floating point are implemented in terms of published standards. While the built-in float type exposes only a modest portion of its capabilities, the decimal module exposes all required parts of the standard. When needed, the programmer has full control over rounding and signal handling. This includes an option to enforce exact arithmetic by using exceptions to block any inexact operations.

• The decimal module was designed to support "without prejudice, both exact unrounded decimal arithmetic (sometimes called fixed-point arithmetic) and rounded floating-point arithmetic." -- excerpt from the decimal arithmetic specification.

The module design is centered around three concepts: the decimal number, the context for arithmetic, and signals.

A decimal number is immutable. It has a sign, coefficient digits, and an exponent. To preserve significance, the coefficient digits do not truncate trailing zeros. Decimals also include special values such as :const:Infinity, :const:-Infinity, and :const:NaN. The standard also differentiates :const:-0 from :const:+0.

The context for arithmetic is an environment specifying precision, rounding rules, limits on exponents, flags indicating the results of operations, and trap enablers which determine whether signals are treated as exceptions. Rounding options include :const:ROUND_CEILING, :const:ROUND_DOWN, :const:ROUND_FLOOR, :const:ROUND_HALF_DOWN, :const:ROUND_HALF_EVEN, :const:ROUND_HALF_UP, :const:ROUND_UP, and :const:ROUND_05UP.

Signals are groups of exceptional conditions arising during the course of computation. Depending on the needs of the application, signals may be ignored, considered as informational, or treated as exceptions. The signals in the decimal module are: :const:Clamped, :const:InvalidOperation, :const:DivisionByZero, :const:Inexact, :const:Rounded, :const:Subnormal, :const:Overflow, :const:Underflow and :const:FloatOperation.

For each signal there is a flag and a trap enabler. When a signal is encountered, its flag is set to one, then, if the trap enabler is set to one, an exception is raised. Flags are sticky, so the user needs to reset them before monitoring a calculation.

## Quick-start Tutorial

The usual start to using decimals is importing the module, viewing the current context with :func:getcontext and, if necessary, setting new values for precision, rounding, or enabled traps:

>>> from decimal import *
>>> getcontext()
Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999,
capitals=1, clamp=0, flags=[], traps=[Overflow, DivisionByZero,
InvalidOperation])

>>> getcontext().prec = 7       # Set a new precision


Decimal instances can be constructed from integers, strings, floats, or tuples. Construction from an integer or a float performs an exact conversion of the value of that integer or float. Decimal numbers include special values such as :const:NaN which stands for "Not a number", positive and negative :const:Infinity, and :const:-0:

>>> getcontext().prec = 28
>>> Decimal(10)
Decimal('10')
>>> Decimal('3.14')
Decimal('3.14')
>>> Decimal(3.14)
Decimal('3.140000000000000124344978758017532527446746826171875')
>>> Decimal((0, (3, 1, 4), -2))
Decimal('3.14')
>>> Decimal(str(2.0 ** 0.5))
Decimal('1.4142135623730951')
>>> Decimal(2) ** Decimal('0.5')
Decimal('1.414213562373095048801688724')
>>> Decimal('NaN')
Decimal('NaN')
>>> Decimal('-Infinity')
Decimal('-Infinity')


If the :exc:FloatOperation signal is trapped, accidental mixing of decimals and floats in constructors or ordering comparisons raises an exception:

>>> c = getcontext()
>>> c.traps[FloatOperation] = True
>>> Decimal(3.14)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
decimal.FloatOperation: [<class 'decimal.FloatOperation'>]
>>> Decimal('3.5') < 3.7
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
decimal.FloatOperation: [<class 'decimal.FloatOperation'>]
>>> Decimal('3.5') == 3.5
True


The significance of a new Decimal is determined solely by the number of digits input. Context precision and rounding only come into play during arithmetic operations.

If the internal limits of the C version are exceeded, constructing a decimal raises :class:InvalidOperation:

>>> Decimal("1e9999999999999999999")
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
decimal.InvalidOperation: [<class 'decimal.InvalidOperation'>]


Decimals interact well with much of the rest of Python. Here is a small decimal floating point flying circus:

And some mathematical functions are also available to Decimal:

>>> getcontext().prec = 28
>>> Decimal(2).sqrt()
Decimal('1.414213562373095048801688724')
>>> Decimal(1).exp()
Decimal('2.718281828459045235360287471')
>>> Decimal('10').ln()
Decimal('2.302585092994045684017991455')
>>> Decimal('10').log10()
Decimal('1')


The :meth:quantize method rounds a number to a fixed exponent. This method is useful for monetary applications that often round results to a fixed number of places:

>>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
Decimal('7.32')
>>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
Decimal('8')


As shown above, the :func:getcontext function accesses the current context and allows the settings to be changed. This approach meets the needs of most applications.

For more advanced work, it may be useful to create alternate contexts using the Context() constructor. To make an alternate active, use the :func:setcontext function.

In accordance with the standard, the :mod:Decimal module provides two ready to use standard contexts, :const:BasicContext and :const:ExtendedContext. The former is especially useful for debugging because many of the traps are enabled:

Contexts also have signal flags for monitoring exceptional conditions encountered during computations. The flags remain set until explicitly cleared, so it is best to clear the flags before each set of monitored computations by using the :meth:clear_flags method.

>>> setcontext(ExtendedContext)
>>> getcontext().clear_flags()
>>> Decimal(355) / Decimal(113)
Decimal('3.14159292')
>>> getcontext()
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999,
capitals=1, clamp=0, flags=[Inexact, Rounded], traps=[])


The flags entry shows that the rational approximation to :const:Pi was rounded (digits beyond the context precision were thrown away) and that the result is inexact (some of the discarded digits were non-zero).

Individual traps are set using the dictionary in the :attr:traps field of a context:

Most programs adjust the current context only once, at the beginning of the program. And, in many applications, data is converted to :class:Decimal with a single cast inside a loop. With context set and decimals created, the bulk of the program manipulates the data no differently than with other Python numeric types.

## Decimal objects

Construct a new :class:Decimal object based from value.

value can be an integer, string, tuple, :class:float, or another :class:Decimal object. If no value is given, returns Decimal('0'). If value is a string, it should conform to the decimal numeric string syntax after leading and trailing whitespace characters are removed:

sign           ::=  '+' | '-'
digit          ::=  '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
indicator      ::=  'e' | 'E'
digits         ::=  digit [digit]...
decimal-part   ::=  digits '.' [digits] | ['.'] digits
exponent-part  ::=  indicator [sign] digits
infinity       ::=  'Infinity' | 'Inf'
nan            ::=  'NaN' [digits] | 'sNaN' [digits]
numeric-value  ::=  decimal-part [exponent-part] | infinity
numeric-string ::=  [sign] numeric-value | [sign] nan


Other Unicode decimal digits are also permitted where digit appears above. These include decimal digits from various other alphabets (for example, Arabic-Indic and Devanāgarī digits) along with the fullwidth digits '\uff10' through '\uff19'.

If value is a :class:tuple, it should have three components, a sign (:const:0 for positive or :const:1 for negative), a :class:tuple of digits, and an integer exponent. For example, Decimal((0, (1, 4, 1, 4), -3)) returns Decimal('1.414').

If value is a :class:float, the binary floating point value is losslessly converted to its exact decimal equivalent. This conversion can often require 53 or more digits of precision. For example, Decimal(float('1.1')) converts to Decimal('1.100000000000000088817841970012523233890533447265625').

The context precision does not affect how many digits are stored. That is determined exclusively by the number of digits in value. For example, Decimal('3.00000') records all five zeros even if the context precision is only three.

The purpose of the context argument is determining what to do if value is a malformed string. If the context traps :const:InvalidOperation, an exception is raised; otherwise, the constructor returns a new Decimal with the value of :const:NaN.

Once constructed, :class:Decimal objects are immutable.

Decimal floating point objects share many properties with the other built-in numeric types such as :class:float and :class:int. All of the usual math operations and special methods apply. Likewise, decimal objects can be copied, pickled, printed, used as dictionary keys, used as set elements, compared, sorted, and coerced to another type (such as :class:float or :class:int).

There are some small differences between arithmetic on Decimal objects and arithmetic on integers and floats. When the remainder operator % is applied to Decimal objects, the sign of the result is the sign of the dividend rather than the sign of the divisor:

>>> (-7) % 4
1
>>> Decimal(-7) % Decimal(4)
Decimal('-3')


The integer division operator // behaves analogously, returning the integer part of the true quotient (truncating towards zero) rather than its floor, so as to preserve the usual identity x == (x // y) * y + x % y:

>>> -7 // 4
-2
>>> Decimal(-7) // Decimal(4)
Decimal('-1')


The % and // operators implement the remainder and divide-integer operations (respectively) as described in the specification.

Decimal objects cannot generally be combined with floats or instances of :class:fractions.Fraction in arithmetic operations: an attempt to add a :class:Decimal to a :class:float, for example, will raise a :exc:TypeError. However, it is possible to use Python's comparison operators to compare a :class:Decimal instance x with another number y. This avoids confusing results when doing equality comparisons between numbers of different types.

In addition to the standard numeric properties, decimal floating point objects also have a number of specialized methods:

### Logical operands

The :meth:logical_and, :meth:logical_invert, :meth:logical_or, and :meth:logical_xor methods expect their arguments to be logical operands. A logical operand is a :class:Decimal instance whose exponent and sign are both zero, and whose digits are all either :const:0 or :const:1.

## Context objects

Contexts are environments for arithmetic operations. They govern precision, set rules for rounding, determine which signals are treated as exceptions, and limit the range for exponents.

Each thread has its own current context which is accessed or changed using the :func:getcontext and :func:setcontext functions:

You can also use the :keyword:with statement and the :func:localcontext function to temporarily change the active context.

New contexts can also be created using the :class:Context constructor described below. In addition, the module provides three pre-made contexts:

This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to :const:ROUND_HALF_UP. All flags are cleared. All traps are enabled (treated as exceptions) except :const:Inexact, :const:Rounded, and :const:Subnormal.

Because many of the traps are enabled, this context is useful for debugging.

This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to :const:ROUND_HALF_EVEN. All flags are cleared. No traps are enabled (so that exceptions are not raised during computations).

Because the traps are disabled, this context is useful for applications that prefer to have result value of :const:NaN or :const:Infinity instead of raising exceptions. This allows an application to complete a run in the presence of conditions that would otherwise halt the program.

This context is used by the :class:Context constructor as a prototype for new contexts. Changing a field (such a precision) has the effect of changing the default for new contexts created by the :class:Context constructor.

This context is most useful in multi-threaded environments. Changing one of the fields before threads are started has the effect of setting system-wide defaults. Changing the fields after threads have started is not recommended as it would require thread synchronization to prevent race conditions.

In single threaded environments, it is preferable to not use this context at all. Instead, simply create contexts explicitly as described below.

The default values are :attr:prec=:const:28, :attr:rounding=:const:ROUND_HALF_EVEN, and enabled traps for :class:Overflow, :class:InvalidOperation, and :class:DivisionByZero.

In addition to the three supplied contexts, new contexts can be created with the :class:Context constructor.

Creates a new context. If a field is not specified or is :const:None, the default values are copied from the :const:DefaultContext. If the flags field is not specified or is :const:None, all flags are cleared.

prec is an integer in the range [:const:1, :const:MAX_PREC] that sets the precision for arithmetic operations in the context.

The rounding option is one of the constants listed in the section Rounding Modes.

The traps and flags fields list any signals to be set. Generally, new contexts should only set traps and leave the flags clear.

The Emin and Emax fields are integers specifying the outer limits allowable for exponents. Emin must be in the range [:const:MIN_EMIN, :const:0], Emax in the range [:const:0, :const:MAX_EMAX].

The capitals field is either :const:0 or :const:1 (the default). If set to :const:1, exponents are printed with a capital :const:E; otherwise, a lowercase :const:e is used: :const:Decimal('6.02e+23').

The clamp field is either :const:0 (the default) or :const:1. If set to :const:1, the exponent e of a :class:Decimal instance representable in this context is strictly limited to the range Emin - prec + 1 <= e <= Emax - prec + 1. If clamp is :const:0 then a weaker condition holds: the adjusted exponent of the :class:Decimal instance is at most Emax. When clamp is :const:1, a large normal number will, where possible, have its exponent reduced and a corresponding number of zeros added to its coefficient, in order to fit the exponent constraints; this preserves the value of the number but loses information about significant trailing zeros. For example:

>>> Context(prec=6, Emax=999, clamp=1).create_decimal('1.23e999')
Decimal('1.23000E+999')


A clamp value of :const:1 allows compatibility with the fixed-width decimal interchange formats specified in IEEE 754.

The :class:Context class defines several general purpose methods as well as a large number of methods for doing arithmetic directly in a given context. In addition, for each of the :class:Decimal methods described above (with the exception of the :meth:adjusted and :meth:as_tuple methods) there is a corresponding :class:Context method. For example, for a :class:Context instance C and :class:Decimal instance x, C.exp(x) is equivalent to x.exp(context=C). Each :class:Context method accepts a Python integer (an instance of :class:int) anywhere that a Decimal instance is accepted.

The usual approach to working with decimals is to create :class:Decimal instances and then apply arithmetic operations which take place within the current context for the active thread. An alternative approach is to use context methods for calculating within a specific context. The methods are similar to those for the :class:Decimal class and are only briefly recounted here.

## Constants

The constants in this section are only relevant for the C module. They are also included in the pure Python version for compatibility.

## Signals

Signals represent conditions that arise during computation. Each corresponds to one context flag and one context trap enabler.

The context flag is set whenever the condition is encountered. After the computation, flags may be checked for informational purposes (for instance, to determine whether a computation was exact). After checking the flags, be sure to clear all flags before starting the next computation.

If the context's trap enabler is set for the signal, then the condition causes a Python exception to be raised. For example, if the :class:DivisionByZero trap is set, then a :exc:DivisionByZero exception is raised upon encountering the condition.

Altered an exponent to fit representation constraints.

Typically, clamping occurs when an exponent falls outside the context's :attr:Emin and :attr:Emax limits. If possible, the exponent is reduced to fit by adding zeros to the coefficient.

Base class for other signals and a subclass of :exc:ArithmeticError.

Signals the division of a non-infinite number by zero.

Can occur with division, modulo division, or when raising a number to a negative power. If this signal is not trapped, returns :const:Infinity or :const:-Infinity with the sign determined by the inputs to the calculation.

Indicates that rounding occurred and the result is not exact.

Signals when non-zero digits were discarded during rounding. The rounded result is returned. The signal flag or trap is used to detect when results are inexact.

An invalid operation was performed.

Indicates that an operation was requested that does not make sense. If not trapped, returns :const:NaN. Possible causes include:

Infinity - Infinity
0 * Infinity
Infinity / Infinity
x % 0
Infinity % x
sqrt(-x) and x > 0
0 ** 0
x ** (non-integer)
x ** Infinity


Numerical overflow.

Indicates the exponent is larger than :attr:Emax after rounding has occurred. If not trapped, the result depends on the rounding mode, either pulling inward to the largest representable finite number or rounding outward to :const:Infinity. In either case, :class:Inexact and :class:Rounded are also signaled.

Rounding occurred though possibly no information was lost.

Signaled whenever rounding discards digits; even if those digits are zero (such as rounding :const:5.00 to :const:5.0). If not trapped, returns the result unchanged. This signal is used to detect loss of significant digits.

Exponent was lower than :attr:Emin prior to rounding.

Occurs when an operation result is subnormal (the exponent is too small). If not trapped, returns the result unchanged.

Numerical underflow with result rounded to zero.

Occurs when a subnormal result is pushed to zero by rounding. :class:Inexact and :class:Subnormal are also signaled.

Enable stricter semantics for mixing floats and Decimals.

If the signal is not trapped (default), mixing floats and Decimals is permitted in the :class:~decimal.Decimal constructor, :meth:~decimal.Context.create_decimal and all comparison operators. Both conversion and comparisons are exact. Any occurrence of a mixed operation is silently recorded by setting :exc:FloatOperation in the context flags. Explicit conversions with :meth:~decimal.Decimal.from_float or :meth:~decimal.Context.create_decimal_from_float do not set the flag.

Otherwise (the signal is trapped), only equality comparisons and explicit conversions are silent. All other mixed operations raise :exc:FloatOperation.

The following table summarizes the hierarchy of signals:

exceptions.ArithmeticError(exceptions.Exception)
DecimalException
Clamped
DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
Inexact
Overflow(Inexact, Rounded)
Underflow(Inexact, Rounded, Subnormal)
InvalidOperation
Rounded
Subnormal
FloatOperation(DecimalException, exceptions.TypeError)


## Floating Point Notes

### Mitigating round-off error with increased precision

The use of decimal floating point eliminates decimal representation error (making it possible to represent :const:0.1 exactly); however, some operations can still incur round-off error when non-zero digits exceed the fixed precision.

The effects of round-off error can be amplified by the addition or subtraction of nearly offsetting quantities resulting in loss of significance. Knuth provides two instructive examples where rounded floating point arithmetic with insufficient precision causes the breakdown of the associative and distributive properties of addition:

The :mod:decimal module makes it possible to restore the identities by expanding the precision sufficiently to avoid loss of significance:

### Special values

The number system for the :mod:decimal module provides special values including :const:NaN, :const:sNaN, :const:-Infinity, :const:Infinity, and two zeros, :const:+0 and :const:-0.

Infinities can be constructed directly with: Decimal('Infinity'). Also, they can arise from dividing by zero when the :exc:DivisionByZero signal is not trapped. Likewise, when the :exc:Overflow signal is not trapped, infinity can result from rounding beyond the limits of the largest representable number.

The infinities are signed (affine) and can be used in arithmetic operations where they get treated as very large, indeterminate numbers. For instance, adding a constant to infinity gives another infinite result.

Some operations are indeterminate and return :const:NaN, or if the :exc:InvalidOperation signal is trapped, raise an exception. For example, 0/0 returns :const:NaN which means "not a number". This variety of :const:NaN is quiet and, once created, will flow through other computations always resulting in another :const:NaN. This behavior can be useful for a series of computations that occasionally have missing inputs --- it allows the calculation to proceed while flagging specific results as invalid.

A variant is :const:sNaN which signals rather than remaining quiet after every operation. This is a useful return value when an invalid result needs to interrupt a calculation for special handling.

The behavior of Python's comparison operators can be a little surprising where a :const:NaN is involved. A test for equality where one of the operands is a quiet or signaling :const:NaN always returns :const:False (even when doing Decimal('NaN')==Decimal('NaN')), while a test for inequality always returns :const:True. An attempt to compare two Decimals using any of the <, <=, > or >= operators will raise the :exc:InvalidOperation signal if either operand is a :const:NaN, and return :const:False if this signal is not trapped. Note that the General Decimal Arithmetic specification does not specify the behavior of direct comparisons; these rules for comparisons involving a :const:NaN were taken from the IEEE 854 standard (see Table 3 in section 5.7). To ensure strict standards-compliance, use the :meth:compare and :meth:compare-signal methods instead.

The signed zeros can result from calculations that underflow. They keep the sign that would have resulted if the calculation had been carried out to greater precision. Since their magnitude is zero, both positive and negative zeros are treated as equal and their sign is informational.

In addition to the two signed zeros which are distinct yet equal, there are various representations of zero with differing precisions yet equivalent in value. This takes a bit of getting used to. For an eye accustomed to normalized floating point representations, it is not immediately obvious that the following calculation returns a value equal to zero:

>>> 1 / Decimal('Infinity')
Decimal('0E-1000026')


The :func:getcontext function accesses a different :class:Context object for each thread. Having separate thread contexts means that threads may make changes (such as getcontext().prec=10) without interfering with other threads.

Likewise, the :func:setcontext function automatically assigns its target to the current thread.

If :func:setcontext has not been called before :func:getcontext, then :func:getcontext will automatically create a new context for use in the current thread.

The new context is copied from a prototype context called DefaultContext. To control the defaults so that each thread will use the same values throughout the application, directly modify the DefaultContext object. This should be done before any threads are started so that there won't be a race condition between threads calling :func:getcontext. For example:

# Set applicationwide defaults for all threads about to be launched
DefaultContext.prec = 12
DefaultContext.rounding = ROUND_DOWN
DefaultContext.traps = ExtendedContext.traps.copy()
DefaultContext.traps[InvalidOperation] = 1
setcontext(DefaultContext)

# Afterwards, the threads can be started
t1.start()
t2.start()
t3.start()
. . .


## Recipes

Here are a few recipes that serve as utility functions and that demonstrate ways to work with the :class:Decimal class:

def moneyfmt(value, places=2, curr='', sep=',', dp='.',
pos='', neg='-', trailneg=''):
"""Convert Decimal to a money formatted string.

places:  required number of places after the decimal point
curr:    optional currency symbol before the sign (may be blank)
sep:     optional grouping separator (comma, period, space, or blank)
dp:      decimal point indicator (comma or period)
only specify as blank when places is zero
pos:     optional sign for positive numbers: '+', space or blank
neg:     optional sign for negative numbers: '-', '(', space or blank
trailneg:optional trailing minus indicator:  '-', ')', space or blank

>>> d = Decimal('-1234567.8901')
>>> moneyfmt(d, curr='$') '-$1,234,567.89'
>>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='-')
'1.234.568-'
>>> moneyfmt(d, curr='$', neg='(', trailneg=')') '($1,234,567.89)'
>>> moneyfmt(Decimal(123456789), sep=' ')
'123 456 789.00'
>>> moneyfmt(Decimal('-0.02'), neg='<', trailneg='>')
'<0.02>'

"""
q = Decimal(10) ** -places      # 2 places --> '0.01'
sign, digits, exp = value.quantize(q).as_tuple()
result = []
digits = list(map(str, digits))
build, next = result.append, digits.pop
if sign:
build(trailneg)
for i in range(places):
build(next() if digits else '0')
if places:
build(dp)
if not digits:
build('0')
i = 0
while digits:
build(next())
i += 1
if i == 3 and digits:
i = 0
build(sep)
build(curr)
build(neg if sign else pos)
return ''.join(reversed(result))

def pi():
"""Compute Pi to the current precision.

>>> print(pi())
3.141592653589793238462643383

"""
getcontext().prec += 2  # extra digits for intermediate steps
three = Decimal(3)      # substitute "three=3.0" for regular floats
lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
while s != lasts:
lasts = s
n, na = n+na, na+8
d, da = d+da, da+32
t = (t * n) / d
s += t
getcontext().prec -= 2
return +s               # unary plus applies the new precision

def exp(x):
"""Return e raised to the power of x.  Result type matches input type.

>>> print(exp(Decimal(1)))
2.718281828459045235360287471
>>> print(exp(Decimal(2)))
7.389056098930650227230427461
>>> print(exp(2.0))
7.38905609893
>>> print(exp(2+0j))
(7.38905609893+0j)

"""
getcontext().prec += 2
i, lasts, s, fact, num = 0, 0, 1, 1, 1
while s != lasts:
lasts = s
i += 1
fact *= i
num *= x
s += num / fact
getcontext().prec -= 2
return +s

def cos(x):
"""Return the cosine of x as measured in radians.

The Taylor series approximation works best for a small value of x.
For larger values, first compute x = x % (2 * pi).

>>> print(cos(Decimal('0.5')))
0.8775825618903727161162815826
>>> print(cos(0.5))
0.87758256189
>>> print(cos(0.5+0j))
(0.87758256189+0j)

"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i-1)
num *= x * x
sign *= -1
s += num / fact * sign
getcontext().prec -= 2
return +s

def sin(x):
"""Return the sine of x as measured in radians.

The Taylor series approximation works best for a small value of x.
For larger values, first compute x = x % (2 * pi).

>>> print(sin(Decimal('0.5')))
0.4794255386042030002732879352
>>> print(sin(0.5))
0.479425538604
>>> print(sin(0.5+0j))
(0.479425538604+0j)

"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i-1)
num *= x * x
sign *= -1
s += num / fact * sign
getcontext().prec -= 2
return +s


## Decimal FAQ

Q. It is cumbersome to type decimal.Decimal('1234.5'). Is there a way to minimize typing when using the interactive interpreter?

1. Some users abbreviate the constructor to just a single letter:

>>> D = decimal.Decimal
>>> D('1.23') + D('3.45')
Decimal('4.68')


Q. In a fixed-point application with two decimal places, some inputs have many places and need to be rounded. Others are not supposed to have excess digits and need to be validated. What methods should be used?

A. The :meth:quantize method rounds to a fixed number of decimal places. If the :const:Inexact trap is set, it is also useful for validation:

>>> TWOPLACES = Decimal(10) ** -2       # same as Decimal('0.01')

>>> # Round to two places
>>> Decimal('3.214').quantize(TWOPLACES)
Decimal('3.21')

>>> # Validate that a number does not exceed two places
>>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact]))
Decimal('3.21')

>>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact]))
Traceback (most recent call last):
...
Inexact: None


Q. Once I have valid two place inputs, how do I maintain that invariant throughout an application?

A. Some operations like addition, subtraction, and multiplication by an integer will automatically preserve fixed point. Others operations, like division and non-integer multiplication, will change the number of decimal places and need to be followed-up with a :meth:quantize step:

>>> a = Decimal('102.72')           # Initial fixed-point values
>>> b = Decimal('3.17')
>>> a + b                           # Addition preserves fixed-point
Decimal('105.89')
>>> a - b
Decimal('99.55')
>>> a * 42                          # So does integer multiplication
Decimal('4314.24')
>>> (a * b).quantize(TWOPLACES)     # Must quantize non-integer multiplication
Decimal('325.62')
>>> (b / a).quantize(TWOPLACES)     # And quantize division
Decimal('0.03')


In developing fixed-point applications, it is convenient to define functions to handle the :meth:quantize step:

>>> def mul(x, y, fp=TWOPLACES):
...     return (x * y).quantize(fp)
>>> def div(x, y, fp=TWOPLACES):
...     return (x / y).quantize(fp)

>>> mul(a, b)                       # Automatically preserve fixed-point
Decimal('325.62')
>>> div(b, a)
Decimal('0.03')


Q. There are many ways to express the same value. The numbers :const:200, :const:200.000, :const:2E2, and :const:.02E+4 all have the same value at various precisions. Is there a way to transform them to a single recognizable canonical value?

A. The :meth:normalize method maps all equivalent values to a single representative:

>>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
>>> [v.normalize() for v in values]
[Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')]


Q. Some decimal values always print with exponential notation. Is there a way to get a non-exponential representation?

A. For some values, exponential notation is the only way to express the number of significant places in the coefficient. For example, expressing :const:5.0E+3 as :const:5000 keeps the value constant but cannot show the original's two-place significance.

If an application does not care about tracking significance, it is easy to remove the exponent and trailing zeroes, losing significance, but keeping the value unchanged:

>>> def remove_exponent(d):
...     return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()

>>> remove_exponent(Decimal('5E+3'))
Decimal('5000')

1. Is there a way to convert a regular float to a :class:Decimal?

A. Yes, any binary floating point number can be exactly expressed as a Decimal though an exact conversion may take more precision than intuition would suggest:

Q. Within a complex calculation, how can I make sure that I haven't gotten a spurious result because of insufficient precision or rounding anomalies.

A. The decimal module makes it easy to test results. A best practice is to re-run calculations using greater precision and with various rounding modes. Widely differing results indicate insufficient precision, rounding mode issues, ill-conditioned inputs, or a numerically unstable algorithm.

Q. I noticed that context precision is applied to the results of operations but not to the inputs. Is there anything to watch out for when mixing values of different precisions?

A. Yes. The principle is that all values are considered to be exact and so is the arithmetic on those values. Only the results are rounded. The advantage for inputs is that "what you type is what you get". A disadvantage is that the results can look odd if you forget that the inputs haven't been rounded:

The solution is either to increase precision or to force rounding of inputs using the unary plus operation:

Alternatively, inputs can be rounded upon creation using the :meth:Context.create_decimal method:

>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
Decimal('1.2345')